# How do I show an impulse response leads to a zero-phase frequency response?

I'm trying to understand how to show that with real coefficients, the phase response of a filter is 0. Here is the impulse response

$h[n] = b_1d[n+1] + b_0d[n] + b_1d[n-1]$

How should I approach showing this?

1) Define $d[n]$ as a sinusoidal

3) Simplify

4) Interpret the results

Hope this isn't too much. Sounds like a homework problem.

Ced

• I'm not sure I expressed myself correctly. h[n] is the impulse response, and so d[n] I'm pretty sure represents impulses ala [1,0,0,...]. I'm supposed to show that it leads to a zero-phase FR when (b0,b1) \in R^2. – user34931 Apr 6 '18 at 22:28
• @user34931, These two links should help: en.wikipedia.org/wiki/Frequency_response and dsp.stackexchange.com/questions/536. – Cedron Dawg Apr 6 '18 at 22:39
• @user34931, Is this a homework problem or are you trying to learn this stuff on your own? Major hint from the dsp link: "For discrete-time systems, this is possible, because you can write any signal x[n] as a sum of scaled and time-shifted Kronecker delta functions:" $$x[n] = \sum_{k=0}^{\infty} x[k] \delta[n - k]$$ From the Wiki article: "It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input." You are asking about the phase. Here is a question and my answer about the magnitude: dsp.stackexchange.com/questions/46510. – Cedron Dawg Apr 6 '18 at 23:15
• Yea this is a homework problem I'm trying to tackle. I understand what an impulse response is, in that a bunch of impulses with scalars, each delayed by different numbers of samples, is essentially an LTI system, (in this case a filter). The part that confuses me is how knowing that this impulse response has a non phase-shifted response. Is it because the IR is symmetrical? – user34931 Apr 6 '18 at 23:21
• @user34931, I suggest you follow my recipe in my answer and see how far you get. Once you have accomplished that, apply the same approach to this definition: $$h[n] = b1*d[n] + b0*d[n-1] + b1*d[n-2]$$ Notice that the phase lag is not zero in this case. Then do a search on "Finite Impulse Response filter with symmetric coefficients" – Cedron Dawg Apr 6 '18 at 23:34

Given an impulse response of the form $h[n]$ for an LTI system, a zero phase freqency response means that the DTFT (discrete-time Fourier transform) $$H(e^{j\omega}) = \sum_{n=-\infty}^{\infty} h[n]e^{-j\omega n}$$ is real and positive.

If it's real but negative, then that's an easily avoidable $\pi$ phase shift and if it's not real but complex then its either linear phase or nonlinear phase...